Boy Born on Tuesday Conditioning Paradox

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Conditioning / Independence.

Your Task
A family has two children. Assume each child is equally likely to be a boy or a girl ($1/2$), and equally likely to be born on any of the 7 days of the week, independently. You are informed that at least one child is a boy born on a Tuesday. What is the conditional probability that both children are boys?

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.